Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 109 / 2 / iv

Table of contents
- Solution iv

Solution

Fix a prime number

p > r

. Distinct members of an intersecting

r

-uniform family have intersection size in

L = {1, 2, \dots, r - 1} \subset F_{p},

whereas every member has size

r \in / L

p

. The Frankl-Wilson theorem with

∣ L ∣ = r - 1

therefore gives

∣ A ∣ \leq (r - 1 n) .

For fixed

r

,

(r - 1 n) = (r - 1 n - 1) \frac{n}{n - r + 1} = (r - 1 n - 1) (1 + O (n^{- 1})) .

This is the asserted asymptotic weakening of the Erdős-Ko-Rado theorem.

Solved by gpt-5.6-sol high.

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